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3 Entanglements and the tube model 47 topological constraints exist. This means, that the number of configurations is somehow restricted, which has a strong influence on the mechanical behavior. On the other hand, the entanglements do not act as full crosslinks, since the constraint is not as severe as in fully fixed monomers. Therefore we may have more degrees of freedom from entanglements than from crosslinks. Thus we can expect that these topological constraints act at two extreme levels. At small deformations we may have restrictions of the sliding of the entanglements, whereas at larger deformations we have restrictions due to the additional tube geometry.

2 Percolation The percolation process describes a more general process for cluster and network formation. In the simple Flory–Stockmayer model introduced above we saw that the main issue is a “connectivity transition” in which a liquid of zero elasticity of functional molecules undergoes a chemical reaction and connects molecules with each other. g. a finite elasticity. We have so far not learned anything about the exponents of the diverging viscosity of the growing elastic modulus. Probably the simplest visualization of the percolation process is given by connecting bonds on a lattice [76, 183].

For the other regime, r ≥ ξ , the self-avoiding correlations do not play a significant role and we can expect chain statistics close to a Gaussian chain. From this naive picture we must conclude that ξ must be a function of the concentration. At this intuitive level we can already deduce one significant concentration, C ∗ , which characterizes the overlap between the chains. If the polymers just overlap, a chain occupies its own volume. Thus we have [63] N N = dν = N 1−νd . 1) d R N This is an important result and we have to note that for large chain lengths N the overlap concentration C ∗ is very small.